That is, the regret bound we get is fully explicit up to a multiplicative constant (which can also be made explicit). Currently we focus on plain (as opposed to catastrophe) and uniform (finite number of hypotheses, uniform prior) DRL, although this result can and should be extended to the catastrophe and/or non-uniform settings.

Appendix A contains the proofs, Appendix B recalls a few lemmas we need from previous essays.

Results

First, we briefly recall some properties of Markov chains.

Definition 1

Consider \({\mathcal{S}}\) a finite set and \({\mathcal{T}}: {\mathcal{S}}{\xrightarrow{\text{k}}}{\mathcal{S}}\). We say that \(k \in {\mathbb{N}}^+\) is a period of \({\mathcal{T}}\) when there is \(s \in {\mathcal{S}}\) an essential state of \({\mathcal{T}}\) (that is, \({\mathcal{T}}^\infty(s \mid s) > 0\)) s.t. \(k\) is its period, i.e. \(k = \gcd {\left\{n \in {\mathbb{N}}^+ \mid {\mathcal{T}}^n(s \mid s) > 0\right\}}\). We denote \({P}_{\mathcal{T}}\) the set of periods of \({\mathcal{T}}\).

The following is a corollary of the Perron-Frobenius theorem which we give without proof. [I believe this is completely standard and would be grateful to get a source for this which treats the reducible case; of course I can produce the proof but it seems redundant.]

For the purpose of this essay, we will use a definition of local sanity slightly stronger than what previously appeared as “Definition 4.” We think this strengthening is not substantial, but also the current analysis can be generalized to the weaker case by adding a term proportional to the 2nd derivative of \({\operatorname{V}}\) (or the 2nd moment of the mixing time). We leave out the details for the time being.

We will use the notation \({\mathcal{A}}_M^\omega(s):=\bigcap_{k\in{\mathbb{N}}} {\mathcal{A}}_M^k(s)\).

Definition 2

Let \(\upsilon = (\mu,r)\) be a universe and \(\epsilon \in (0,1)\). A policy \(\sigma\) is said to be locally \(\epsilon\)-sane for \(\upsilon\) when there are \(M\), \(S\) and \({\mathcal{U}}_M \subseteq {\mathcal{S}}_M\) (the set of uncorrupt states) s.t. \(\upsilon\) is an \({\mathcal{O}}\)-realization of \(M\) with state function \(S\), \(S({\boldsymbol{\lambda}}) \in {\mathcal{U}}_M\) and for any \(h \in \operatorname{hdom}{\mu}\), if \(S(h) \in {\mathcal{U}}_M\) then

Recall that for any MPD \(M\), there is \(\gamma_M\in(0,1)\) s.t. for any \(\gamma\in[\gamma_M,1)\), \(a \in {\mathcal{A}}_M^\omega(s)\) if and only if \({\operatorname{Q}}_M(s,a,\gamma)={\operatorname{V}}_M(s,\gamma)\).

Assuming w.l.o.g. that all uncorrupt states are reachable from \(S^k({\boldsymbol{\lambda}})\), \(\pi^k\) is guaranteed to exist thanks to condition iii of Definition 2 (if some uncorrupt state is unreachable, we can consider it to be corrupt.) Let \(F_k\in(0,\infty)\), \(\lambda_k\in(0,1)\) and \({P}_k \subseteq {\mathbb{N}}^+\) be as in Proposition 1, for the Markov chain \({\mathcal{T}}_{M^k\pi^k}: {\mathcal{U}}^k {\xrightarrow{\text{k}}}{\mathcal{U}}^k\). Then, there is an \(\bar{{\mathcal{I}}}\)-policy \(\pi^*\) s.t. for any \(k \in [N]\)

Appendix A

The proof of Theorem 1 mostly repeats the proof of the previous “Theorem 1”, except that we keep track of the bounds more carefully.

Proof of Theorem 1

Fix \(\eta\in\left(0,N^{-1}\right)\) and \(T \in {\mathbb{N}}^+\). Denote \(\nu^k:=\bar{\mu}^k\left[\sigma^k S^k\right]\). To avoid cumbersome notation, whenever \(M^k\) should appear a subscript, we will replace it by \(k\). Let \((\Omega,P \in \Delta\Omega)\) be a probability space. Let \(K: \Omega \rightarrow [N]\) be a random variable and the following be stochastic processes

\[{Z}_n,\tilde{{Z}}_n: \Omega \rightarrow \Delta[N]\]

\[{J}_n: \Omega \rightarrow [N]\]

\[\Psi_n: \Omega \rightarrow {\mathcal{A}}\]

\[A_n: \Omega \rightarrow {\bar{{\mathcal{A}}}}\]

\[\Theta_n: \Omega \rightarrow {\bar{{\mathcal{O}}}}\]

We also define \(A\Theta_{:n}: \Omega \rightarrow {{\overline{{\mathcal{A}}\times {\mathcal{O}}}}^*}\) by

\[A\Theta_{:n}:= A_0\Theta_0A_1\Theta_1 \ldots A_{n-1}\Theta_{n-1}\]

(The following conditions on \(A\) and \(\Theta\) imply that the range of the above is indeed in \({{\overline{{\mathcal{A}}\times {\mathcal{O}}}}^*}\).) Let \({\mathcal{D}}\) and \({\mathcal{D}}^{!k}\) be as in Proposition B.1 (the form of the bound we are proving allows assuming w.l.o.g. that \(\epsilon < \frac{1}{{\left\vert {\mathcal{A}}\right\vert}}\)). By condition iii of Definition 2, there is \(\pi^k : \operatorname{hdom}{\mu^k} \rightarrow {\mathcal{A}}\) s.t. for any \(h \in \operatorname{hdom}{\mu^k}\), if \(S^k(h) \in {\mathcal{U}}^k\) then

That is, we perform Thompson sampling at time intervals of size \(T\), moderated by the delegation routine \({\mathcal{D}}\), and discard from our belief state hypotheses whose probability is below \(\eta\) and hypotheses sampling which resulted in recommending “unsafe” actions i.e. actions that \({\mathcal{D}}\) refused to perform.

In order to prove \(\pi^*\) has the desired property, we will define the stochastic processes \({Z}^!\), \(\tilde{{Z}}^!\), \({J}^!\), \(\Psi^!\), \(A^!\) and \(\Theta^!\), each process of the same type as its shriekless counterpart (thus \(\Omega\) is constructed to accommodate them). These processes are required to satisfy the following:

Given any \({{\bar{{\mathcal{I}}}}}\)-policy \(\pi\) and \({\mathcal{I}}\)-policy \(\sigma\) we define \(\alpha_{\sigma\pi}: {({\mathcal{A}}\times {\mathcal{O}})^*}{\xrightarrow{\text{k}}}{{\overline{{\mathcal{A}}\times {\mathcal{O}}}}^*}\) by

Here, the \({\mathcal{I}}\)-policy \(\pi^{*k}_n\) is defined as \(\pi^*_n\) in Proposition B.2, with \(\pi^k\) in place of \(\pi^*\). We also define the \({{\bar{{\mathcal{I}}}}}\)-policies \(\pi^{!k}_n\) and \(\pi^{!!k}_n\) by

Here, the factor of 2 comes from the difference between the equations for \(Z_n\) and \(Z^!_n\) (we can construct and intermediate policy between \(\pi^*\) and \(\pi^{?k}\) and use the triangle inequality for \({\operatorname{d}_{\text{tv}}}\)). We conclude

Without loss of generality, we can assume this to be consistent with the assumption \(\eta < \frac{1}{N}\) since the bound contains a term of \(N^2\eta\) anyway. Similarly, the second term in the bound implies we can assume that \(1 - (1-\alpha)^T \ll 1\) and therefore \(1 - (1-\alpha)^T = O(T\alpha)\). Finally, note that assumption ii implies that the expression we round to get \(T\) is \(\geq 1\). We get

The following is a simple modification of what appeared there as “Proposition B.2” (the corresponding modification of the proof is trivial and we leave it out.)

Proposition B.2

Consider some \(\gamma\in(0,1)\), \(\tau\in(0,\infty)\), \(T\in{\mathbb{N}}^+\), a universe \(\upsilon=(\mu,r)\) that is an \({\mathcal{O}}\)-realization of \(M\) with state function \(S\), some \(\pi^*: \operatorname{hdom}{\mu} \rightarrow {\mathcal{A}}\) and some \(\pi^0: \operatorname{hdom}{\mu} {\xrightarrow{\text{k}}}{\mathcal{A}}\). Assume that \(\gamma \geq \gamma_M\). For any \(n \in {\mathbb{N}}\), let \(\pi^*_n\) be an \({\mathcal{I}}\)-policy s.t. for any \(h \in \operatorname{hdom}{\mu}\)